scholarly journals Generalized second-order derivatives of convex functions in reflexive Banach spaces

1992 ◽  
Vol 334 (1) ◽  
pp. 281-301 ◽  
Author(s):  
Chi Ngoc Do
Keyword(s):  
Banach Spaces ◽  
Convex Functions ◽  
Second Order ◽  
Reflexive Banach Spaces ◽  
Derivatives Of

scholarly journals Generalised second-order derivatives of convex functions in reflexive Banach spaces

1995 ◽  
Vol 51 (1) ◽  
pp. 55-72 ◽  
Author(s):  
James Louis Ndoutoume ◽  
Michel Théra
Keyword(s):  
Banach Spaces ◽  
Convex Functions ◽  
Quadratic Forms ◽  
Second Order ◽  
Second Derivative ◽  
Reflexive Banach Spaces ◽  
Finite Dimensional ◽  
Derivatives Of

Generalised second-order derivatives introduced by Rockafellar in the finite dimensional setting are extended to convex functions defined on reflexive Banach spaces. Our approach is based on the characterisation of convex generalised quadratic forms defined in reflexive Banach spaces, from the graph of the associated subdifferentials. The main result which is obtained is the exhibition of a particular generalised Hessian when the function admits a generalised second derivative. Some properties of the generalised second derivative are pointed out along with further justifications of the concept.

scholarly journals Simpson’s Second-Type Inequalities for Co-Ordinated Convex Functions and Applications for Cubature Formulas

2022 ◽  
Vol 6 (1) ◽  
pp. 33
Author(s):  
Sabah Iftikhar ◽  
Samet Erden ◽  
Muhammad Aamir Ali ◽  
Jamel Baili ◽  
Hijaz Ahmad
Keyword(s):  
Convex Functions ◽  
Cubature Formula ◽  
Applied Mathematics ◽  
Second Order ◽  
Absolute Value ◽  
Partial Derivatives ◽  
Cubature Formulas ◽  
Inequality Theory ◽  
Type Integral ◽  
Derivatives Of

Inequality theory has attracted considerable attention from scientists because it can be used in many fields. In particular, Hermite–Hadamard and Simpson inequalities based on convex functions have become a cornerstone in pure and applied mathematics. We deal with Simpson’s second-type inequalities based on coordinated convex functions in this work. In this paper, we first introduce Simpson’s second-type integral inequalities for two-variable functions whose second-order partial derivatives in modulus are convex on the coordinates. In addition, similar results are acquired by considering that powers of the absolute value of second-order partial derivatives of these two-variable functions are convex on the coordinates. Finally, some applications for Simpson’s 3/8 cubature formula are given.

A characterization of essentially strictly convex functions on reflexive Banach spaces

2012 ◽  
Vol 75 (3) ◽  
pp. 1617-1622 ◽  
Author(s):  
Michel Volle ◽  
Jean-Baptiste Hiriart-Urruty
Keyword(s):  
Banach Spaces ◽  
Convex Functions ◽  
Reflexive Banach Spaces ◽  
Strictly Convex

scholarly journals New Hermite-Hadamard type inequalities for m and (α, m)-convex functions on the coordinates via generalized fractional integrals

2021 ◽  
Vol 40 (6) ◽  
pp. 1449-1472
Author(s):  
Seth Kermausuor
Keyword(s):  
Convex Functions ◽  
Type Inequality ◽  
Second Order ◽  
Fractional Integrals ◽  
Absolute Value ◽  
Partial Derivatives ◽  
Independent Variables ◽  
Functions Of Two Variables ◽  
Derivatives Of ◽  
Two Independent Variables

In this paper, we obtained a new Hermite-Hadamard type inequality for functions of two independent variables that are m-convex on the coordinates via some generalized Katugampola type fractional integrals. We also established a new identity involving the second order mixed partial derivatives of functions of two independent variables via the generalized Katugampola fractional integrals. Using the identity, we established some new Hermite-Hadamard type inequalities for functions whose second order mixed partial derivatives in absolute value at some powers are (α, m)-convex on the coordinates. Our results are extensions of some earlier results in the literature for functions of two variables.

Second-Order Gâteaux Differentiable Bump Functions and Approximations in Banach Spaces

1993 ◽  
Vol 45 (3) ◽  
pp. 612-625 ◽  
Author(s):  
D. McLaughlin ◽  
R. Poliquin ◽  
J. Vanderwerff ◽  
V. Zizler
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Convex Functions ◽  
Second Order ◽  
Separable Spaces

AbstractIn this paper we study approximations of convex functions by twice Gâteaux differentiate convex functions. We prove that convex functions (respectively norms) can be approximated by twice Gâteaux differentiate convex functions (respectively norms) in separable Banach spaces which have the Radon-Nikody m property and admit twice Gâteaux differentiable bump functions. New characterizations of spaces isomorphic to Hilbert spaces are shown. Locally uniformly rotund norms that are limits of Ck-smooth norms are constructed in separable spaces which admit Ck-smooth norms.

scholarly journals Characterization of Reflexivity by Convex Functions

2016 ◽  
Vol 2016 ◽  
pp. 1-4
Author(s):  
Zhenghua Luo ◽  
Qingjin Cheng
Keyword(s):  
Banach Spaces ◽  
Convex Functions ◽  
Convexity Property ◽  
Reflexive Banach Spaces

A new convexity property of convex functions is introduced. This property provides, in particular, a characterization of the class of reflexive Banach spaces.

scholarly journals New Tour on the Subdifferential of Supremum via Finite Sums and Suprema

2021 ◽  
Author(s):  
A. Hantoute ◽  
M. A. López-Cerdá
Keyword(s):  
Banach Spaces ◽  
Convex Functions ◽  
Normal Cone ◽  
Countable Family ◽  
Weighted Sums ◽  
Reflexive Banach Spaces ◽  
Finite Dimensional ◽  
Finite Sums ◽  
Effective Domain ◽  
The Relationship

AbstractThis paper provides new characterizations for the subdifferential of the pointwise supremum of an arbitrary family of convex functions. The main feature of our approach is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of it) does not appear in our formulas. Another aspect of our analysis is that it emphasizes the relationship with the subdifferential of the supremum of finite subfamilies, or equivalently, finite weighted sums. Some specific results are given in the setting of reflexive Banach spaces, showing that the subdifferential of the supremum can be reduced to the supremum of a countable family.

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